3.15.24 \(\int \frac {b+2 c x}{(d+e x)^{3/2} (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=354 \[ \frac {\sqrt {2} \sqrt {c} \left (b e \left (\sqrt {b^2-4 a c}+b\right )-2 c \left (d \sqrt {b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {2} \sqrt {c} \left (2 c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac {2 (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )} \]

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Rubi [A]  time = 0.70, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {828, 826, 1166, 208} \begin {gather*} \frac {\sqrt {2} \sqrt {c} \left (b e \left (\sqrt {b^2-4 a c}+b\right )-2 c \left (d \sqrt {b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {2} \sqrt {c} \left (2 c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac {2 (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/((d + e*x)^(3/2)*(a + b*x + c*x^2)),x]

[Out]

(2*(2*c*d - b*e))/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[c]*(b*(b + Sqrt[b^2 - 4*a*c])*e - 2*
c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*
e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[2]*Sqrt[c]*(b
*(b - Sqrt[b^2 - 4*a*c])*e + 2*c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2
*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e
+ a*e^2))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx &=\frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\int \frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c d^2-b d e+a e^2}\\ &=\frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {2 \operatorname {Subst}\left (\int \frac {-c d (2 c d-b e)+e \left (b c d-b^2 e+2 a c e\right )+c (2 c d-b e) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c d^2-b d e+a e^2}\\ &=\frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\left (2 \left (\frac {1}{2} c (2 c d-b e)-\frac {-c (2 c d-b e) (-2 c d+b e)+2 c \left (-c d (2 c d-b e)+e \left (b c d-b^2 e+2 a c e\right )\right )}{2 \sqrt {b^2-4 a c} e}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{c d^2-b d e+a e^2}+\frac {\left (2 \left (\frac {1}{2} c (2 c d-b e)+\frac {-c (2 c d-b e) (-2 c d+b e)+2 c \left (-c d (2 c d-b e)+e \left (b c d-b^2 e+2 a c e\right )\right )}{2 \sqrt {b^2-4 a c} e}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{c d^2-b d e+a e^2}\\ &=\frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {c} \left (b \left (b+\sqrt {b^2-4 a c}\right ) e-2 c \left (\sqrt {b^2-4 a c} d+2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {2} \sqrt {c} \left (b \left (b-\sqrt {b^2-4 a c}\right ) e+c \left (2 \sqrt {b^2-4 a c} d-4 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.50, size = 317, normalized size = 0.90 \begin {gather*} \frac {2 \left (\frac {\sqrt {c} \left (2 c \left (d \sqrt {b^2-4 a c}+2 a e\right )-b e \left (\sqrt {b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}+\frac {\sqrt {c} \left (2 c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {b e-2 c d}{\sqrt {d+e x}}\right )}{e (b d-a e)-c d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)/((d + e*x)^(3/2)*(a + b*x + c*x^2)),x]

[Out]

(2*((-2*c*d + b*e)/Sqrt[d + e*x] + (Sqrt[c]*(-(b*(b + Sqrt[b^2 - 4*a*c])*e) + 2*c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e
))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c
]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[c]*(b*(b - Sqrt[b^2 - 4*a*c])*e + 2*c*(Sqrt[b^2 - 4*a*c]*d
- 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2
- 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/(-(c*d^2) + e*(b*d - a*e))

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IntegrateAlgebraic [A]  time = 1.26, size = 416, normalized size = 1.18 \begin {gather*} -\frac {\left (2 \sqrt {2} c^{3/2} d \sqrt {b^2-4 a c}-\sqrt {2} b \sqrt {c} e \sqrt {b^2-4 a c}+4 \sqrt {2} a c^{3/2} e-\sqrt {2} b^2 \sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-e \sqrt {b^2-4 a c}+b e-2 c d}}\right )}{\sqrt {b^2-4 a c} \sqrt {-e \sqrt {b^2-4 a c}+b e-2 c d} \left (-a e^2+b d e-c d^2\right )}-\frac {\left (2 \sqrt {2} c^{3/2} d \sqrt {b^2-4 a c}-\sqrt {2} b \sqrt {c} e \sqrt {b^2-4 a c}-4 \sqrt {2} a c^{3/2} e+\sqrt {2} b^2 \sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}+b e-2 c d}}\right )}{\sqrt {b^2-4 a c} \sqrt {e \sqrt {b^2-4 a c}+b e-2 c d} \left (-a e^2+b d e-c d^2\right )}+\frac {2 (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + 2*c*x)/((d + e*x)^(3/2)*(a + b*x + c*x^2)),x]

[Out]

(2*(2*c*d - b*e))/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) - ((2*Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*d - Sqrt[2]*
b^2*Sqrt[c]*e + 4*Sqrt[2]*a*c^(3/2)*e - Sqrt[2]*b*Sqrt[c]*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d
+ e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]
*(-(c*d^2) + b*d*e - a*e^2)) - ((2*Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*d + Sqrt[2]*b^2*Sqrt[c]*e - 4*Sqrt[2]*a*c
^(3/2)*e - Sqrt[2]*b*Sqrt[c]*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + S
qrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2))

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fricas [B]  time = 0.72, size = 8557, normalized size = 24.17

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/2*(sqrt(2)*(c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 + a*e^3)*x)*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(
b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c +
a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 1
8*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 +
4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d
*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b
^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c
^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c
^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*
e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)
)*log(sqrt(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e + (11*b^2*c^2 - 8*a*c^3)*d^2*e^2 - (5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^
4 - 3*a*b^2*c + 2*a^2*c^2)*e^4 - (2*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*
(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*
b^2 + 2*a^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 -
22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2
*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e
^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c
^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*
c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9
+ 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a
*b*c)*e^3 + (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d
^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3
*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4
*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2
*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*
c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5
+ 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4
*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^
2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)) - 4*(3*c^4*d^2 - 3*b*c^3*d*e + (b
^2*c^2 - a*c^3)*e^2)*sqrt(e*x + d)) - sqrt(2)*(c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 + a*e^3)*x)*sqrt
((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (c^3*d^6 - 3*b*c^2*d^5*e - 3*a
^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((
9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^
2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*
d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c
^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 +
(b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 +
15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^
2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3
*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4))*log(-sqrt(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e + (11*b^2*c^2 - 8*a*c^3)*d^2*e^2 -
(5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^4 - 3*a*b^2*c + 2*a^2*c^2)*e^4 - (2*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*
(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*
b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*
a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*
e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^1
2 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^
4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3
)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 1
0*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b
^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c +
a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18
*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4
*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*
e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^
2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^
2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^
2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e
- 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4))
- 4*(3*c^4*d^2 - 3*b*c^3*d*e + (b^2*c^2 - a*c^3)*e^2)*sqrt(e*x + d)) + sqrt(2)*(c*d^3 - b*d^2*e + a*d*e^2 + (
c*d^2*e - b*d*e^2 + a*e^3)*x)*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^
3 - (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 +
3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*
c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a
^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*
d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a
*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^
2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^
3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*
c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4))*log(sqrt(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e +
(11*b^2*c^2 - 8*a*c^3)*d^2*e^2 - (5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^4 - 3*a*b^2*c + 2*a^2*c^2)*e^4 + (2*c^4*d^7
- 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b
^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*
a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*
c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*
d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 1
5*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^
4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4
*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))*sqrt(
(2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^
2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9
*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2
*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d
^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^
4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 +
(b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 1
5*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2
*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*
e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)) - 4*(3*c^4*d^2 - 3*b*c^3*d*e + (b^2*c^2 - a*c^3)*e^2)*sqrt(e*x + d)) - sqrt(
2)*(c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 + a*e^3)*x)*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*
a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^
4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3
- 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^
3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^
6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a
^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^
3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^
8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*
b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4))*log(-sqr
t(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e + (11*b^2*c^2 - 8*a*c^3)*d^2*e^2 - (5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^4 - 3*a*b
^2*c + 2*a^2*c^2)*e^4 + (2*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c +
3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a
^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*
c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*
a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(
2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a
^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^
3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^
4*b^2 + 2*a^5*c)*d^2*e^10)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3
- (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 +
3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c
^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^
2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d
^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*
b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2
*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3
*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c
^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)) - 4*(3*c^4*d^2 - 3*b*c^3*d*e + (b^2*c^2 -
a*c^3)*e^2)*sqrt(e*x + d)) + 4*(2*c*d - b*e)*sqrt(e*x + d))/(c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 +
a*e^3)*x)

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giac [B]  time = 1.00, size = 1366, normalized size = 3.86

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

2*(2*c*d - b*e)/((c*d^2 - b*d*e + a*e^2)*sqrt(x*e + d)) + 1/4*((c*d^2*e - b*d*e^2 + a*e^3)^2*sqrt(-4*c^2*d + 2
*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*sqrt(b^2 - 4*a*c)*c*d - sqrt(b^2 - 4*a*c)*b*e) - 2*(2*c^3*d^4 - 4*b*c^2*d^3
*e + 3*b^2*c*d^2*e^2 - b^3*d*e^3 + (a*b^2 - 2*a^2*c)*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs
(c*d^2*e - b*d*e^2 + a*e^3) + (2*sqrt(b^2 - 4*a*c)*c^3*d^5*e^2 - 5*sqrt(b^2 - 4*a*c)*b*c^2*d^4*e^3 + 4*(b^2*c
+ a*c^2)*sqrt(b^2 - 4*a*c)*d^3*e^4 - sqrt(b^2 - 4*a*c)*a^2*b*e^7 - (b^3 + 6*a*b*c)*sqrt(b^2 - 4*a*c)*d^2*e^5 +
2*(a*b^2 + a^2*c)*sqrt(b^2 - 4*a*c)*d*e^6)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/
2)*sqrt(x*e + d)/sqrt(-(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3 + sqrt((2*c^2*d^3 - 3*b*c*
d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3)^2 - 4*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*
d*e^3 + a^2*e^4)*(c^2*d^2 - b*c*d*e + a*c*e^2)))/(c^2*d^2 - b*c*d*e + a*c*e^2)))/((c^3*d^6 - 3*b*c^2*d^5*e + 3
*(b^2*c + a*c^2)*d^4*e^2 - 3*a^2*b*d*e^5 - (b^3 + 6*a*b*c)*d^3*e^3 + a^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^4)*abs(
c*d^2*e - b*d*e^2 + a*e^3)*abs(c)) - 1/4*((c*d^2*e - b*d*e^2 + a*e^3)^2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*
a*c)*c)*e)*(2*sqrt(b^2 - 4*a*c)*c*d - sqrt(b^2 - 4*a*c)*b*e) + 2*(2*c^3*d^4 - 4*b*c^2*d^3*e + 3*b^2*c*d^2*e^2
- b^3*d*e^3 + (a*b^2 - 2*a^2*c)*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c*d^2*e - b*d*e^2 +
a*e^3) + (2*sqrt(b^2 - 4*a*c)*c^3*d^5*e^2 - 5*sqrt(b^2 - 4*a*c)*b*c^2*d^4*e^3 + 4*(b^2*c + a*c^2)*sqrt(b^2 - 4
*a*c)*d^3*e^4 - sqrt(b^2 - 4*a*c)*a^2*b*e^7 - (b^3 + 6*a*b*c)*sqrt(b^2 - 4*a*c)*d^2*e^5 + 2*(a*b^2 + a^2*c)*sq
rt(b^2 - 4*a*c)*d*e^6)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt
(-(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3 - sqrt((2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2
*a*c*d*e^2 - a*b*e^3)^2 - 4*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*(c^2
*d^2 - b*c*d*e + a*c*e^2)))/(c^2*d^2 - b*c*d*e + a*c*e^2)))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*(b^2*c + a*c^2)*d^4*
e^2 - 3*a^2*b*d*e^5 - (b^3 + 6*a*b*c)*d^3*e^3 + a^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^4)*abs(c*d^2*e - b*d*e^2 + a
*e^3)*abs(c))

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maple [B]  time = 0.11, size = 927, normalized size = 2.62 \begin {gather*} -\frac {4 \sqrt {2}\, a \,c^{2} e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {4 \sqrt {2}\, a \,c^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {\sqrt {2}\, b^{2} c \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {\sqrt {2}\, b^{2} c \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {\sqrt {2}\, b c e \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, b c e \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {2 \sqrt {2}\, c^{2} d \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}+\frac {2 \sqrt {2}\, c^{2} d \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {2 b e}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}}+\frac {4 c d}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {e x +d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x)

[Out]

-4/(a*e^2-b*d*e+c*d^2)*c^2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*ar
ctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*e^2+1/(a*e^2-b*d*e+c*d^2)*c/(
-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/
((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*e^2+1/(a*e^2-b*d*e+c*d^2)*c*2^(1/2)/((-b*e+2*c*d+(-(4*a
*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*
b*e-2/(a*e^2-b*d*e+c*d^2)*c^2*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^
(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d-4/(a*e^2-b*d*e+c*d^2)*c^2/(-(4*a*c-b^2)*e^2)^(1/2)*
2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*
e^2)^(1/2))*c)^(1/2)*c)*a*e^2+1/(a*e^2-b*d*e+c*d^2)*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^
2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*e^2
-1/(a*e^2-b*d*e+c*d^2)*c*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/(
(b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*e+2/(a*e^2-b*d*e+c*d^2)*c^2*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^
2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d-2/(a*
e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)*b*e+4/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)*c*d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, c x + b}{{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate((2*c*x + b)/((c*x^2 + b*x + a)*(e*x + d)^(3/2)), x)

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mupad [B]  time = 9.40, size = 33147, normalized size = 93.64

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)/((d + e*x)^(3/2)*(a + b*x + c*x^2)),x)

[Out]

(4*c*d - 2*b*e + 2^(1/2)*a*e^2*atan(-((2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*
e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^
(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2
+ 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)
*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^
4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^
4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 1
28*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5
*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5
*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*
e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^
(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2
+ 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e
^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 -
128*a^4*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 +
192*b^5*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 - (2^(1/2)*(-(b^3*e^3 - 2*c
^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2
*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b
^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d
^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*
a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16
*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56
*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d
^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e
^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 7
20*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^
4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480
*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12))/2 + 1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a
^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2*e^10 + 608*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4
*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d^2*e^10 - 320*a*b*c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 -
576*a*b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d
^3*e^9 - 768*a^2*b*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a
^4*b^3*c^3*d*e^11))/2)*1i)/2 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a
*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) -
3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2
*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*
c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d
^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e
^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*
b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 +
40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*
e^9 - 128*a^3*b^3*c^4*d*e^9) - (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a
*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) -
3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2
*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64
*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 - 128*a^4
*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 + 192*b^5
*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3
+ b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*
b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*
e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e -
6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3
*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^
8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^
3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8
+ 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80
*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^
2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 -
1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*
c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12))/2 + 1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a^2*b^4*
c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2*e^10 + 608*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4*c^3*d^
2*e^10 + 48*a^4*b^2*c^4*d^2*e^10 - 320*a*b*c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 - 576*a*
b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d^3*e^9
- 768*a^2*b*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a^4*b^3*
c^3*d*e^11))/2)*1i)/2)/(8*a*b^7*c^2*e^9 - 128*a^4*b*c^5*e^9 + 256*a^4*c^6*d*e^8 - 8*b^8*c^2*d*e^8 - 64*a^4*c^5
*e^9*(b^2 - 4*a*c)^(1/2) - 72*a^2*b^5*c^3*e^9 + 192*a^3*b^3*c^4*e^9 - 768*a^2*c^8*d^5*e^4 - 512*a^3*c^7*d^3*e^
6 - 48*b^4*c^6*d^5*e^4 + 120*b^5*c^5*d^4*e^5 - 112*b^6*c^4*d^3*e^6 + 48*b^7*c^3*d^2*e^7 - 56*a^2*b^4*c^3*e^9*(
b^2 - 4*a*c)^(1/2) + 112*a^3*b^2*c^4*e^9*(b^2 - 4*a*c)^(1/2) + 64*a^2*c^7*d^4*e^5*(b^2 - 4*a*c)^(1/2) + 192*a^
3*c^6*d^2*e^7*(b^2 - 4*a*c)^(1/2) + 48*b^2*c^7*d^6*e^3*(b^2 - 4*a*c)^(1/2) - 144*b^3*c^6*d^5*e^4*(b^2 - 4*a*c)
^(1/2) + 184*b^4*c^5*d^4*e^5*(b^2 - 4*a*c)^(1/2) - 128*b^5*c^4*d^3*e^6*(b^2 - 4*a*c)^(1/2) + 48*b^6*c^3*d^2*e^
7*(b^2 - 4*a*c)^(1/2) - 1536*a^2*b^2*c^6*d^3*e^6 + 384*a^2*b^3*c^5*d^2*e^7 + 32*a*b^6*c^3*d*e^8 + 8*a*b^6*c^2*
e^9*(b^2 - 4*a*c)^(1/2) - 192*a*c^8*d^6*e^3*(b^2 - 4*a*c)^(1/2) - 8*b^7*c^2*d*e^8*(b^2 - 4*a*c)^(1/2) + 384*a*
b^2*c^7*d^5*e^4 - 960*a*b^3*c^6*d^4*e^5 + 864*a*b^4*c^5*d^3*e^6 - 336*a*b^5*c^4*d^2*e^7 + 1920*a^2*b*c^7*d^4*e
^5 + 144*a^2*b^4*c^4*d*e^8 + 768*a^3*b*c^6*d^2*e^7 - 640*a^3*b^2*c^5*d*e^8 + 576*a*b*c^7*d^5*e^4*(b^2 - 4*a*c)
^(1/2) + 16*a*b^5*c^3*d*e^8*(b^2 - 4*a*c)^(1/2) - 192*a^3*b*c^5*d*e^8*(b^2 - 4*a*c)^(1/2) - 752*a*b^2*c^6*d^4*
e^5*(b^2 - 4*a*c)^(1/2) + 544*a*b^3*c^5*d^3*e^6*(b^2 - 4*a*c)^(1/2) - 192*a*b^4*c^4*d^2*e^7*(b^2 - 4*a*c)^(1/2
) - 128*a^2*b*c^6*d^3*e^6*(b^2 - 4*a*c)^(1/2) + 112*a^2*b^3*c^4*d*e^8*(b^2 - 4*a*c)^(1/2) - 48*a^2*b^2*c^5*d^2
*e^7*(b^2 - 4*a*c)^(1/2)))*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 -
4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*
(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 +
3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*1i + 2^(1/2)*a*e^2*a
tan(-((2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2
) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c
)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*
e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*
e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e
^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 +
480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2*e^8 - 256*a*b
*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48
*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e
^9) + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2
) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c
)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*
e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4
*c^2*e^12 + 48*a^5*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 - 128*a^4*c^6*d^4*e^8 - 192*a^5*c^
5*d^2*e^10 - 16*b^2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 + 192*b^5*c^5*d^7*e^5 - 128*b^6*c^
4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 - (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(
1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e
*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3
*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(
d + e*x)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d
^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d
^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5
*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9
- 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1
200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3
*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 -
2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^
2*c^3*d*e^12))/2 + 1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*
c^2*d^2*e^10 + 608*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d
^2*e^10 - 320*a*b*c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 - 576*a*b^3*c^6*d^7*e^5 + 192*a*b
^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d^3*e^9 - 768*a^2*b*c^7*d^7*e^5 -
384*a^3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a^4*b^3*c^3*d*e^11))/2)*1i)/2 + (
2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a
*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2)
)/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3
*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 -
8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64
*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2
*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^
7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*
c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) - (
2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a
*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2)
)/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3
*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^
12 + 48*a^5*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 - 128*a^4*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e
^10 - 16*b^2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 + 192*b^5*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e
^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) -
3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 -
4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*
d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x
)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4
+ 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3
- 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 +
2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a
^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4
*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^
8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a
^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d
*e^12))/2 + 1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2
*e^10 + 608*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d^2*e^10
- 320*a*b*c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 - 576*a*b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*
d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d^3*e^9 - 768*a^2*b*c^7*d^7*e^5 - 384*a^
3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a^4*b^3*c^3*d*e^11))/2)*1i)/2)/(8*a*b^7*
c^2*e^9 - 128*a^4*b*c^5*e^9 + 256*a^4*c^6*d*e^8 - 8*b^8*c^2*d*e^8 + 64*a^4*c^5*e^9*(b^2 - 4*a*c)^(1/2) - 72*a^
2*b^5*c^3*e^9 + 192*a^3*b^3*c^4*e^9 - 768*a^2*c^8*d^5*e^4 - 512*a^3*c^7*d^3*e^6 - 48*b^4*c^6*d^5*e^4 + 120*b^5
*c^5*d^4*e^5 - 112*b^6*c^4*d^3*e^6 + 48*b^7*c^3*d^2*e^7 + 56*a^2*b^4*c^3*e^9*(b^2 - 4*a*c)^(1/2) - 112*a^3*b^2
*c^4*e^9*(b^2 - 4*a*c)^(1/2) - 64*a^2*c^7*d^4*e^5*(b^2 - 4*a*c)^(1/2) - 192*a^3*c^6*d^2*e^7*(b^2 - 4*a*c)^(1/2
) - 48*b^2*c^7*d^6*e^3*(b^2 - 4*a*c)^(1/2) + 144*b^3*c^6*d^5*e^4*(b^2 - 4*a*c)^(1/2) - 184*b^4*c^5*d^4*e^5*(b^
2 - 4*a*c)^(1/2) + 128*b^5*c^4*d^3*e^6*(b^2 - 4*a*c)^(1/2) - 48*b^6*c^3*d^2*e^7*(b^2 - 4*a*c)^(1/2) - 1536*a^2
*b^2*c^6*d^3*e^6 + 384*a^2*b^3*c^5*d^2*e^7 + 32*a*b^6*c^3*d*e^8 - 8*a*b^6*c^2*e^9*(b^2 - 4*a*c)^(1/2) + 192*a*
c^8*d^6*e^3*(b^2 - 4*a*c)^(1/2) + 8*b^7*c^2*d*e^8*(b^2 - 4*a*c)^(1/2) + 384*a*b^2*c^7*d^5*e^4 - 960*a*b^3*c^6*
d^4*e^5 + 864*a*b^4*c^5*d^3*e^6 - 336*a*b^5*c^4*d^2*e^7 + 1920*a^2*b*c^7*d^4*e^5 + 144*a^2*b^4*c^4*d*e^8 + 768
*a^3*b*c^6*d^2*e^7 - 640*a^3*b^2*c^5*d*e^8 - 576*a*b*c^7*d^5*e^4*(b^2 - 4*a*c)^(1/2) - 16*a*b^5*c^3*d*e^8*(b^2
- 4*a*c)^(1/2) + 192*a^3*b*c^5*d*e^8*(b^2 - 4*a*c)^(1/2) + 752*a*b^2*c^6*d^4*e^5*(b^2 - 4*a*c)^(1/2) - 544*a*
b^3*c^5*d^3*e^6*(b^2 - 4*a*c)^(1/2) + 192*a*b^4*c^4*d^2*e^7*(b^2 - 4*a*c)^(1/2) + 128*a^2*b*c^6*d^3*e^6*(b^2 -
4*a*c)^(1/2) - 112*a^2*b^3*c^4*d*e^8*(b^2 - 4*a*c)^(1/2) + 48*a^2*b^2*c^5*d^2*e^7*(b^2 - 4*a*c)^(1/2)))*(-(b^
3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 +
3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 +
c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5
- 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*1i + 2^(1/2)*c*d^2*atan(-((2^(1/2)*(-(b^3*e^3 - 2*c^
3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*
e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^
3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^
5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a
^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^
4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^
2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e
^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^
3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) + (2^(1/2)*(-(b^3*e^3 - 2*c^
3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*
e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^
3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^
5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^3*e^12
+ 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 - 128*a^4*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d^10*e^2
+ 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 + 192*b^5*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 -
8*b^8*c^2*d^4*e^8 - (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^
2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e
^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4
+ 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*(64*a*c^9*d^11*e^
2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 6
40*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 +
240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6
- 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 272
0*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a
^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6
*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a
^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12))/2 + 1248*a^2*b^2*
c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2*e^10 + 608*a^3*b^2*c^5*d
^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d^2*e^10 - 320*a*b*c^8*d^9*e^3 +
192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 - 576*a*b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^
5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d^3*e^9 - 768*a^2*b*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5*e^7 + 32*a^3*
b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a^4*b^3*c^3*d*e^11))/2)*1i)/2 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 +
b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b
^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e
^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6
*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*
c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d
^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c
^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 38
4*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e
^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) - (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 +
b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b
^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e
^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6
*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^3*e^12 + 192*a
^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 - 128*a^4*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d^10*e^2 + 80*b
^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 + 192*b^5*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*
c^2*d^4*e^8 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a
*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2
- 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^
2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*(64*a*c^9*d^11*e^2 - 32*
a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*
c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5
*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*
a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b
^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*
c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 -
208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^
5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12))/2 + 1248*a^2*b^2*c^6*d^6
*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2*e^10 + 608*a^3*b^2*c^5*d^4*e^8
- 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d^2*e^10 - 320*a*b*c^8*d^9*e^3 + 192*a^5
*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 - 576*a*b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 -
112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d^3*e^9 - 768*a^2*b*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2
*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a^4*b^3*c^3*d*e^11))/2)*1i)/2)/(8*a*b^7*c^2*e^9 - 128*a^4*b*c^5*e^9 + 25
6*a^4*c^6*d*e^8 - 8*b^8*c^2*d*e^8 - 64*a^4*c^5*e^9*(b^2 - 4*a*c)^(1/2) - 72*a^2*b^5*c^3*e^9 + 192*a^3*b^3*c^4*
e^9 - 768*a^2*c^8*d^5*e^4 - 512*a^3*c^7*d^3*e^6 - 48*b^4*c^6*d^5*e^4 + 120*b^5*c^5*d^4*e^5 - 112*b^6*c^4*d^3*e
^6 + 48*b^7*c^3*d^2*e^7 - 56*a^2*b^4*c^3*e^9*(b^2 - 4*a*c)^(1/2) + 112*a^3*b^2*c^4*e^9*(b^2 - 4*a*c)^(1/2) + 6
4*a^2*c^7*d^4*e^5*(b^2 - 4*a*c)^(1/2) + 192*a^3*c^6*d^2*e^7*(b^2 - 4*a*c)^(1/2) + 48*b^2*c^7*d^6*e^3*(b^2 - 4*
a*c)^(1/2) - 144*b^3*c^6*d^5*e^4*(b^2 - 4*a*c)^(1/2) + 184*b^4*c^5*d^4*e^5*(b^2 - 4*a*c)^(1/2) - 128*b^5*c^4*d
^3*e^6*(b^2 - 4*a*c)^(1/2) + 48*b^6*c^3*d^2*e^7*(b^2 - 4*a*c)^(1/2) - 1536*a^2*b^2*c^6*d^3*e^6 + 384*a^2*b^3*c
^5*d^2*e^7 + 32*a*b^6*c^3*d*e^8 + 8*a*b^6*c^2*e^9*(b^2 - 4*a*c)^(1/2) - 192*a*c^8*d^6*e^3*(b^2 - 4*a*c)^(1/2)
- 8*b^7*c^2*d*e^8*(b^2 - 4*a*c)^(1/2) + 384*a*b^2*c^7*d^5*e^4 - 960*a*b^3*c^6*d^4*e^5 + 864*a*b^4*c^5*d^3*e^6
- 336*a*b^5*c^4*d^2*e^7 + 1920*a^2*b*c^7*d^4*e^5 + 144*a^2*b^4*c^4*d*e^8 + 768*a^3*b*c^6*d^2*e^7 - 640*a^3*b^2
*c^5*d*e^8 + 576*a*b*c^7*d^5*e^4*(b^2 - 4*a*c)^(1/2) + 16*a*b^5*c^3*d*e^8*(b^2 - 4*a*c)^(1/2) - 192*a^3*b*c^5*
d*e^8*(b^2 - 4*a*c)^(1/2) - 752*a*b^2*c^6*d^4*e^5*(b^2 - 4*a*c)^(1/2) + 544*a*b^3*c^5*d^3*e^6*(b^2 - 4*a*c)^(1
/2) - 192*a*b^4*c^4*d^2*e^7*(b^2 - 4*a*c)^(1/2) - 128*a^2*b*c^6*d^3*e^6*(b^2 - 4*a*c)^(1/2) + 112*a^2*b^3*c^4*
d*e^8*(b^2 - 4*a*c)^(1/2) - 48*a^2*b^2*c^5*d^2*e^7*(b^2 - 4*a*c)^(1/2)))*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2
- 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 +
3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*
d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^
3))^(1/2)*(d + e*x)^(1/2)*1i + 2^(1/2)*c*d^2*atan(-((2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1
/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*
(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*
a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((
d + e*x)^(1/2)*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^
6*e^4 - 128*a^4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e
^5 - 40*b^6*c^4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c
^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256
*a*b^3*c^6*d^5*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^
5 + 24*a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1
/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*
(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*
a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(6
4*a*c^9*d^10*e^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*
c^7*d^6*e^6 - 128*a^4*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*
c^6*d^8*e^4 + 192*b^5*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 - (2^(1/2)*(-
(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2
+ 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6
+ c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e
^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c
^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5
*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^
4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240
*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3
*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^
8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 4
0*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^
2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12))/2 + 1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d
^5*e^7 + 432*a^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2*e^10 + 608*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9
+ 192*a^3*b^4*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d^2*e^10 - 320*a*b*c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2
*c^7*d^8*e^4 - 576*a*b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 +
32*a*b^7*c^2*d^3*e^9 - 768*a^2*b*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d
^3*e^9 - 176*a^4*b^3*c^3*d*e^11))/2)*1i)/2 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3
*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 -
4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d
^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x
)^(1/2)*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 -
128*a^4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40
*b^6*c^4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*
e^8 + 128*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*
c^6*d^5*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*
a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) - (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3
*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 -
4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d
^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9
*d^10*e^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6
*e^6 - 128*a^4*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8
*e^4 + 192*b^5*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 + (2^(1/2)*(-(b^3*e^
3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*
c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*
d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*
b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^
12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^
10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e
^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^
4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^
4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*
e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7
*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^1
2 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12))/2 + 1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7
+ 432*a^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2*e^10 + 608*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*
a^3*b^4*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d^2*e^10 - 320*a*b*c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^
8*e^4 - 576*a*b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^
7*c^2*d^3*e^9 - 768*a^2*b*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9
- 176*a^4*b^3*c^3*d*e^11))/2)*1i)/2)/(8*a*b^7*c^2*e^9 - 128*a^4*b*c^5*e^9 + 256*a^4*c^6*d*e^8 - 8*b^8*c^2*d*e^
8 + 64*a^4*c^5*e^9*(b^2 - 4*a*c)^(1/2) - 72*a^2*b^5*c^3*e^9 + 192*a^3*b^3*c^4*e^9 - 768*a^2*c^8*d^5*e^4 - 512*
a^3*c^7*d^3*e^6 - 48*b^4*c^6*d^5*e^4 + 120*b^5*c^5*d^4*e^5 - 112*b^6*c^4*d^3*e^6 + 48*b^7*c^3*d^2*e^7 + 56*a^2
*b^4*c^3*e^9*(b^2 - 4*a*c)^(1/2) - 112*a^3*b^2*c^4*e^9*(b^2 - 4*a*c)^(1/2) - 64*a^2*c^7*d^4*e^5*(b^2 - 4*a*c)^
(1/2) - 192*a^3*c^6*d^2*e^7*(b^2 - 4*a*c)^(1/2) - 48*b^2*c^7*d^6*e^3*(b^2 - 4*a*c)^(1/2) + 144*b^3*c^6*d^5*e^4
*(b^2 - 4*a*c)^(1/2) - 184*b^4*c^5*d^4*e^5*(b^2 - 4*a*c)^(1/2) + 128*b^5*c^4*d^3*e^6*(b^2 - 4*a*c)^(1/2) - 48*
b^6*c^3*d^2*e^7*(b^2 - 4*a*c)^(1/2) - 1536*a^2*b^2*c^6*d^3*e^6 + 384*a^2*b^3*c^5*d^2*e^7 + 32*a*b^6*c^3*d*e^8
- 8*a*b^6*c^2*e^9*(b^2 - 4*a*c)^(1/2) + 192*a*c^8*d^6*e^3*(b^2 - 4*a*c)^(1/2) + 8*b^7*c^2*d*e^8*(b^2 - 4*a*c)^
(1/2) + 384*a*b^2*c^7*d^5*e^4 - 960*a*b^3*c^6*d^4*e^5 + 864*a*b^4*c^5*d^3*e^6 - 336*a*b^5*c^4*d^2*e^7 + 1920*a
^2*b*c^7*d^4*e^5 + 144*a^2*b^4*c^4*d*e^8 + 768*a^3*b*c^6*d^2*e^7 - 640*a^3*b^2*c^5*d*e^8 - 576*a*b*c^7*d^5*e^4
*(b^2 - 4*a*c)^(1/2) - 16*a*b^5*c^3*d*e^8*(b^2 - 4*a*c)^(1/2) + 192*a^3*b*c^5*d*e^8*(b^2 - 4*a*c)^(1/2) + 752*
a*b^2*c^6*d^4*e^5*(b^2 - 4*a*c)^(1/2) - 544*a*b^3*c^5*d^3*e^6*(b^2 - 4*a*c)^(1/2) + 192*a*b^4*c^4*d^2*e^7*(b^2
- 4*a*c)^(1/2) + 128*a^2*b*c^6*d^3*e^6*(b^2 - 4*a*c)^(1/2) - 112*a^2*b^3*c^4*d*e^8*(b^2 - 4*a*c)^(1/2) + 48*a
^2*b^2*c^5*d^2*e^7*(b^2 - 4*a*c)^(1/2)))*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 +
a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2)
+ 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^
2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*1i - 2
^(1/2)*b*d*e*atan(-((2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2
- 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^
2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4
+ 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*c^9*d^8*e^2
- 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d^2*e^8 - 16
*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e^6 + 8*b^7*
c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2
*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 + 40*a*b^4*c^
5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*e^9 - 128*a
^3*b^3*c^4*d*e^9) + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2
- 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^
2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4
+ 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64*a^6*c^4*e^
12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 - 128*a^4*c^6*d^4*e^
8 - 192*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 + 192*b^5*c^5*d^7*e^
5 - 128*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 - (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(
b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2
+ 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b
^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3
*e^3))^(1/2)*(d + e*x)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 +
320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2
+ 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 -
8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b
^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^
2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e
^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b
*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^1
1 + 144*a^5*b^2*c^3*d*e^12))/2 + 1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a^2*b^4*c^4*d^4*e^8
- 48*a^2*b^6*c^2*d^2*e^10 + 608*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4*c^3*d^2*e^10 + 48
*a^4*b^2*c^4*d^2*e^10 - 320*a*b*c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 - 576*a*b^3*c^6*d^7
*e^5 + 192*a*b^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d^3*e^9 - 768*a^2*b
*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a^4*b^3*c^3*d*e^11)
)/2)*1i)/2 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*
c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2
- 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2
*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*c^9*d^8*e^2 - 64*a
^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d^2*e^8 - 16*b^2*c^
8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e^6 + 8*b^7*c^3*d^3
*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2*e^8 -
256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 + 40*a*b^4*c^5*d^4*e
^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*
c^4*d*e^9) - (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*
c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2
- 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2
*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64*a^6*c^4*e^12 - 8*
a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 - 128*a^4*c^6*d^4*e^8 - 192
*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 + 192*b^5*c^5*d^7*e^5 - 128
*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4
*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^
2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*
e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^
(1/2)*(d + e*x)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^
2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^
3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*
c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*
d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e
^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 60
0*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^
8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144
*a^5*b^2*c^3*d*e^12))/2 + 1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a^2*b^4*c^4*d^4*e^8 - 48*a
^2*b^6*c^2*d^2*e^10 + 608*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4*c^3*d^2*e^10 + 48*a^4*b^
2*c^4*d^2*e^10 - 320*a*b*c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 - 576*a*b^3*c^6*d^7*e^5 +
192*a*b^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d^3*e^9 - 768*a^2*b*c^7*d^
7*e^5 - 384*a^3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a^4*b^3*c^3*d*e^11))/2)*1i
)/2)/(8*a*b^7*c^2*e^9 - 128*a^4*b*c^5*e^9 + 256*a^4*c^6*d*e^8 - 8*b^8*c^2*d*e^8 - 64*a^4*c^5*e^9*(b^2 - 4*a*c)
^(1/2) - 72*a^2*b^5*c^3*e^9 + 192*a^3*b^3*c^4*e^9 - 768*a^2*c^8*d^5*e^4 - 512*a^3*c^7*d^3*e^6 - 48*b^4*c^6*d^5
*e^4 + 120*b^5*c^5*d^4*e^5 - 112*b^6*c^4*d^3*e^6 + 48*b^7*c^3*d^2*e^7 - 56*a^2*b^4*c^3*e^9*(b^2 - 4*a*c)^(1/2)
+ 112*a^3*b^2*c^4*e^9*(b^2 - 4*a*c)^(1/2) + 64*a^2*c^7*d^4*e^5*(b^2 - 4*a*c)^(1/2) + 192*a^3*c^6*d^2*e^7*(b^2
- 4*a*c)^(1/2) + 48*b^2*c^7*d^6*e^3*(b^2 - 4*a*c)^(1/2) - 144*b^3*c^6*d^5*e^4*(b^2 - 4*a*c)^(1/2) + 184*b^4*c
^5*d^4*e^5*(b^2 - 4*a*c)^(1/2) - 128*b^5*c^4*d^3*e^6*(b^2 - 4*a*c)^(1/2) + 48*b^6*c^3*d^2*e^7*(b^2 - 4*a*c)^(1
/2) - 1536*a^2*b^2*c^6*d^3*e^6 + 384*a^2*b^3*c^5*d^2*e^7 + 32*a*b^6*c^3*d*e^8 + 8*a*b^6*c^2*e^9*(b^2 - 4*a*c)^
(1/2) - 192*a*c^8*d^6*e^3*(b^2 - 4*a*c)^(1/2) - 8*b^7*c^2*d*e^8*(b^2 - 4*a*c)^(1/2) + 384*a*b^2*c^7*d^5*e^4 -
960*a*b^3*c^6*d^4*e^5 + 864*a*b^4*c^5*d^3*e^6 - 336*a*b^5*c^4*d^2*e^7 + 1920*a^2*b*c^7*d^4*e^5 + 144*a^2*b^4*c
^4*d*e^8 + 768*a^3*b*c^6*d^2*e^7 - 640*a^3*b^2*c^5*d*e^8 + 576*a*b*c^7*d^5*e^4*(b^2 - 4*a*c)^(1/2) + 16*a*b^5*
c^3*d*e^8*(b^2 - 4*a*c)^(1/2) - 192*a^3*b*c^5*d*e^8*(b^2 - 4*a*c)^(1/2) - 752*a*b^2*c^6*d^4*e^5*(b^2 - 4*a*c)^
(1/2) + 544*a*b^3*c^5*d^3*e^6*(b^2 - 4*a*c)^(1/2) - 192*a*b^4*c^4*d^2*e^7*(b^2 - 4*a*c)^(1/2) - 128*a^2*b*c^6*
d^3*e^6*(b^2 - 4*a*c)^(1/2) + 112*a^2*b^3*c^4*d*e^8*(b^2 - 4*a*c)^(1/2) - 48*a^2*b^2*c^5*d^2*e^7*(b^2 - 4*a*c)
^(1/2)))*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*
a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2
))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 -
3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*1i - 2^(1/2)*b*d*e*atan(-((2^(1/2)*(-(
b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2
+ 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6
+ c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^
5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c
^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^
7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^
4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128
*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7
- 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) + (2^(1/2)*(-(
b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2
+ 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6
+ c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^
5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5
*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 - 128*a^4*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e^10 - 16*b^
2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 + 192*b^5*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e^6 + 48*b^7
*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 - (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3
+ a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/
2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3
*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*(64
*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c
^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*
c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^
2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*
d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^
3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336
*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^
6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12))/2 +
1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2*e^10 + 608
*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d^2*e^10 - 320*a*b*
c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 - 576*a*b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*d^6*e^6 + 9
6*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d^3*e^9 - 768*a^2*b*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5
*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a^4*b^3*c^3*d*e^11))/2)*1i)/2 + (2^(1/2)*(-(b^3*e^3
- 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c
^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d
^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b
*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10
+ 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 -
104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 -
320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*
c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a
*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) - (2^(1/2)*(-(b^3*e^3
- 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c
^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d
^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b
*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^
3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 - 128*a^4*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d
^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 + 192*b^5*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^
5*e^7 - 8*b^8*c^2*d^4*e^8 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*
e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*
b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*
d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*(64*a*c^9*
d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*
e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9
*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d
^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^1
0 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10
- 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 720*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*
c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 -
1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12))/2 + 1248*a
^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2*e^10 + 608*a^3*b^
2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d^2*e^10 - 320*a*b*c^8*d^9
*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 - 576*a*b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*d^6*e^6 + 96*a*b^5
*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d^3*e^9 - 768*a^2*b*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5*e^7 +
32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a^4*b^3*c^3*d*e^11))/2)*1i)/2)/(8*a*b^7*c^2*e^9 - 128*a^4*
b*c^5*e^9 + 256*a^4*c^6*d*e^8 - 8*b^8*c^2*d*e^8 + 64*a^4*c^5*e^9*(b^2 - 4*a*c)^(1/2) - 72*a^2*b^5*c^3*e^9 + 19
2*a^3*b^3*c^4*e^9 - 768*a^2*c^8*d^5*e^4 - 512*a^3*c^7*d^3*e^6 - 48*b^4*c^6*d^5*e^4 + 120*b^5*c^5*d^4*e^5 - 112
*b^6*c^4*d^3*e^6 + 48*b^7*c^3*d^2*e^7 + 56*a^2*b^4*c^3*e^9*(b^2 - 4*a*c)^(1/2) - 112*a^3*b^2*c^4*e^9*(b^2 - 4*
a*c)^(1/2) - 64*a^2*c^7*d^4*e^5*(b^2 - 4*a*c)^(1/2) - 192*a^3*c^6*d^2*e^7*(b^2 - 4*a*c)^(1/2) - 48*b^2*c^7*d^6
*e^3*(b^2 - 4*a*c)^(1/2) + 144*b^3*c^6*d^5*e^4*(b^2 - 4*a*c)^(1/2) - 184*b^4*c^5*d^4*e^5*(b^2 - 4*a*c)^(1/2) +
128*b^5*c^4*d^3*e^6*(b^2 - 4*a*c)^(1/2) - 48*b^6*c^3*d^2*e^7*(b^2 - 4*a*c)^(1/2) - 1536*a^2*b^2*c^6*d^3*e^6 +
384*a^2*b^3*c^5*d^2*e^7 + 32*a*b^6*c^3*d*e^8 - 8*a*b^6*c^2*e^9*(b^2 - 4*a*c)^(1/2) + 192*a*c^8*d^6*e^3*(b^2 -
4*a*c)^(1/2) + 8*b^7*c^2*d*e^8*(b^2 - 4*a*c)^(1/2) + 384*a*b^2*c^7*d^5*e^4 - 960*a*b^3*c^6*d^4*e^5 + 864*a*b^
4*c^5*d^3*e^6 - 336*a*b^5*c^4*d^2*e^7 + 1920*a^2*b*c^7*d^4*e^5 + 144*a^2*b^4*c^4*d*e^8 + 768*a^3*b*c^6*d^2*e^7
- 640*a^3*b^2*c^5*d*e^8 - 576*a*b*c^7*d^5*e^4*(b^2 - 4*a*c)^(1/2) - 16*a*b^5*c^3*d*e^8*(b^2 - 4*a*c)^(1/2) +
192*a^3*b*c^5*d*e^8*(b^2 - 4*a*c)^(1/2) + 752*a*b^2*c^6*d^4*e^5*(b^2 - 4*a*c)^(1/2) - 544*a*b^3*c^5*d^3*e^6*(b
^2 - 4*a*c)^(1/2) + 192*a*b^4*c^4*d^2*e^7*(b^2 - 4*a*c)^(1/2) + 128*a^2*b*c^6*d^3*e^6*(b^2 - 4*a*c)^(1/2) - 11
2*a^2*b^3*c^4*d*e^8*(b^2 - 4*a*c)^(1/2) + 48*a^2*b^2*c^5*d^2*e^7*(b^2 - 4*a*c)^(1/2)))*(-(b^3*e^3 - 2*c^3*d^3
- b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*
b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*
e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e -
6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*1i)/((d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)**(3/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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